--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/HahnBanach/Bounds.thy Tue Dec 30 11:10:01 2008 +0100
@@ -0,0 +1,82 @@
+(* Title: HOL/Real/HahnBanach/Bounds.thy
+ Author: Gertrud Bauer, TU Munich
+*)
+
+header {* Bounds *}
+
+theory Bounds
+imports Main ContNotDenum
+begin
+
+locale lub =
+ fixes A and x
+ assumes least [intro?]: "(\<And>a. a \<in> A \<Longrightarrow> a \<le> b) \<Longrightarrow> x \<le> b"
+ and upper [intro?]: "a \<in> A \<Longrightarrow> a \<le> x"
+
+lemmas [elim?] = lub.least lub.upper
+
+definition
+ the_lub :: "'a::order set \<Rightarrow> 'a" where
+ "the_lub A = The (lub A)"
+
+notation (xsymbols)
+ the_lub ("\<Squnion>_" [90] 90)
+
+lemma the_lub_equality [elim?]:
+ assumes "lub A x"
+ shows "\<Squnion>A = (x::'a::order)"
+proof -
+ interpret lub A x by fact
+ show ?thesis
+ proof (unfold the_lub_def)
+ from `lub A x` show "The (lub A) = x"
+ proof
+ fix x' assume lub': "lub A x'"
+ show "x' = x"
+ proof (rule order_antisym)
+ from lub' show "x' \<le> x"
+ proof
+ fix a assume "a \<in> A"
+ then show "a \<le> x" ..
+ qed
+ show "x \<le> x'"
+ proof
+ fix a assume "a \<in> A"
+ with lub' show "a \<le> x'" ..
+ qed
+ qed
+ qed
+ qed
+qed
+
+lemma the_lubI_ex:
+ assumes ex: "\<exists>x. lub A x"
+ shows "lub A (\<Squnion>A)"
+proof -
+ from ex obtain x where x: "lub A x" ..
+ also from x have [symmetric]: "\<Squnion>A = x" ..
+ finally show ?thesis .
+qed
+
+lemma lub_compat: "lub A x = isLub UNIV A x"
+proof -
+ have "isUb UNIV A = (\<lambda>x. A *<= x \<and> x \<in> UNIV)"
+ by (rule ext) (simp only: isUb_def)
+ then show ?thesis
+ by (simp only: lub_def isLub_def leastP_def setge_def setle_def) blast
+qed
+
+lemma real_complete:
+ fixes A :: "real set"
+ assumes nonempty: "\<exists>a. a \<in> A"
+ and ex_upper: "\<exists>y. \<forall>a \<in> A. a \<le> y"
+ shows "\<exists>x. lub A x"
+proof -
+ from ex_upper have "\<exists>y. isUb UNIV A y"
+ unfolding isUb_def setle_def by blast
+ with nonempty have "\<exists>x. isLub UNIV A x"
+ by (rule reals_complete)
+ then show ?thesis by (simp only: lub_compat)
+qed
+
+end